Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x-2y &= 4 \\ -3x+2y &= -8\end{align*}$
Solution: Begin by moving the $x$ -term in the second equation to the right side of the equation. $2y = 3x-8$ Divide both sides by $2$ to isolate $y$ $y = {\dfrac{3}{2}x - 4}$ Substitute this expression for $y$ in the first equation. $-5x-2({\dfrac{3}{2}x - 4}) = 4$ $-5x - 3x + 8 = 4$ Simplify by combining terms, then solve for $x$ $-8x + 8 = 4$ $-8x = -4$ $x = \dfrac{1}{2}$ Substitute $\dfrac{1}{2}$ for $x$ back into the top equation. $-5( \dfrac{1}{2})-2y = 4$ $-\dfrac{5}{2}-2y = 4$ $-2y = \dfrac{13}{2}$ $y = -\dfrac{13}{4}$ The solution is $\enspace x = \dfrac{1}{2}, \enspace y = -\dfrac{13}{4}$.